Animated Solution for Mathematics - Definite Integration: The integral ${\int }_0^{\pi }\frac{8xdx}{4{\cos }^{2}x+{\sin }^{2}x}$ is equal to

The integral $I={\int }_0^{\pi }\frac{8x}{4{\cos }^{2}x+{\sin }^{2}x}dx$ appears intimidating due to the presence of $x$ in the numerator. This term prevents the direct application of standard trigonometric identities.

In the context of JEE Advanced, such problems are designed to be solved using symmetry properties.

The Master Key

Whenever an unwanted $x$ is multiplied by trigonometric functions in a definite integral, the King's Property is the most effective tool. It states that:

Applying this by replacing $x$ with $\pi -x$, we get:

Since ${\cos }^2(\pi -x)={\cos }^{2}x$ and ${\sin }^2(\pi -x)={\sin }^{2}x$, the denominator remains unchanged. Adding the two expressions for $I$ causes the $x$ terms to cancel out:

Simplifying the expression, we obtain:

The integrand is symmetric about $x=\frac{\pi }{2}$. We apply the Queen's Property, ${\int }_0^{2a}f(x)dx=2{\int }_0^{a}f(x)dx$, to halve the upper limit:

To handle the even powers of sine and cosine, we divide both the numerator and the denominator by ${\cos }^{2}x$. This transforms the integral into:

We perform the substitution $t=\tan x$, which implies $dt={\sec }^{2}xdx$. As $x$ ranges from $0$ to $\frac{\pi }{2}$, $t$ ranges from $0$ to $\infty$.

The integral becomes:

Using the standard integral formula $\int \frac{dx}{a^2+x^2}=\frac{1}{a}{\tan }^{-1}(\frac{x}{a})$, we evaluate:

Evaluating the limits, we find:

The complexity of the problem was merely a mask for a beautiful, underlying symmetry.